One of the difficulties in dealing with the subject of tongues, especially from a cessationist's point of view, is explaining the existence of the gift in Apostolic times. Until the canon was complete, the existence of revelatory gifts made some sense; but why the gift of tongues? A common answer to this question is that tongues were used in a missionary context: at the inception of the church, it was expedient that people from many different language groups be exposed to the gospel in a timely fashion; therefore God gave tongues to those engaged in missions so they could communicate with people whose languages they had not learned.
This "missionary use of tongues" is a popularly held position; even early Pentecostals believed it, thinking that the gift they had been given would enable them to do missions work without learning foreign languages. The attempt actually to do so generally ended in failure, which is one reason why modern tongues were derided as being something different than the biblical precedent. In fact, the alleged missionary use of tongues has absolutely no biblical precedent at all, other than a doubtful interpretation of Acts chapter 2, in which people heard the believers in the Upper Room speaking in their own native languages (Acts 2:6-11). However, these were people who had traveled to Jerusalem to attend the "Feast of Weeks" (Lev. 23:15-16); it is unlikely that they would have been unable to understand either Aramaic or Greek. There is no suggestion in the passage that Peter addressed the crowd in a divinely-inspired language, or that anyone had any difficulty understanding him. It is more likely that God gave the believers the native dialects of the visitors in order to validate the gift, rather than to give the gospel to those who otherwise wouldn't understand it.
1 Corinthians 14
Paul's instructions regarding the gifts of tongues and prophecy in 1 Corinthians 14 make clear that tongues, even in the way he was instructing that the gift be used, were not generally used for the purpose of communicating the gospel to people of another language. This is not to say that God can't use tongues in that way, and there is anecdotal evidence that He has done so. It is simply to say that there is no biblical precedent for this use. It is not the primary reason why tongues was given in the first place.
Cessationists strongly emphasize that the overall point of 1 Corinthians 14 is to diminish its importance as compared with the gift of prophecy (often combined with the idea that "prophecy" is to be identified with preaching). That is, in fact, an important theme in the chapter, but it is not the only one. Paul also wants to give guidelines on the proper use of tongues in the context of the church. Paul's first statement directly contradicts that tongues was only used for the "missionary use." "Anyone who speaks in a tongue does not speak to men but to God. Indeed, no one understands him; he utters mysteries with his spirit" (v. 2)
This single verse is so damaging to the "missionary" theory that John MacArthur tried to argue that since Θεω here is anarthrous, that the Corinthians have been speaking to "a god," which is to say, a false god--the tongues-speaking Corinthians are talking to demons! (I nearly drove off the road the first time I heard him actually say this on the radio.) Of course, this argument is virtually identical with the one Jehovah's Witnesses use on John 1:1; the lack of an article here is meaningless. Paul is quite clearly saying that speaking in tongues is not directed toward people nor is it understood by them; it is rather directed toward God. If tongues were for missionary purposes, they would be directed toward people, who would understand them--that would be their function. But Paul quite clearly states that their function is to speak "mysteries" to God. So much for those who claim that there is no biblical precedent for a "private prayer language."
Coming up: more on 1 Corinthians 14.